Category:Definitions/Borel Sigma-Algebras

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This category contains definitions related to Borel Sigma-Algebras.
Related results can be found in Category:Borel Sigma-Algebras.


Topological Space

Let $\struct {S, \tau}$ be a topological space

The Borel sigma-algebra $\map \BB {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$.

That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.


Metric Space

Let $\struct {S, \norm {\, \cdot \,} }$ be a metric space.

The Borel sigma-algebra (or $\sigma$-algebra) on $\struct {S, \norm {\, \cdot \,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset S$.

By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on a topological space.